Engineering Transactions, 23, 3, pp. 393-421, 1975

Extremum and Variational Principles in Plasticity

H. Lippmann

Extrema of functions (functionals) W are for numerical reasons frequently expressed in the weaker variational form dW = 0. The relationship between both is discussed. In most theories W denotes somewhat like work, and dW = 0 the principle of virtual work. In the Sec. 2 the variational and extremum principles for rigid plastic materials are considered. The classical dual Harr-von Karman-Sadowski-Phillips-Hill upper and lower bound principles are given in general form which is not restricted to specific boundary conditions, or to incompressibility, rate-independence, homogenity, or istotropy. They have become one of the strongest tools for applying theory to practical problems. This is illustrated by a series of examples taken from structural mechanics, metal forming technology and soil mechanics to show also some recently studies features concerning surface fraction, action of volume forces, and volume compression, or extension, respectively. The Sec. 3 concerns the static problems for elastic-plastic material. Starting from the Cotterill-Castigliano principles for elasticity, Prager, Hodge, Greenberg and Bauer developed similar principles for perfectly-plastic or strain-hardening materials. There are, however, only few numerical applications. Recently it has even been shown that utmost caution is necessary to avoid systematic errors. In the Sec. 4 the rate-dependent of dynamic plasticity is discussed. Besides general principles of mechanics like Hamilton’s principle there are a few specific ones partly related to the theorem of work and energy which allow to estimate the magnitude of total deformation, of other quantities. The Sec. 5 is devoted to some generalizations and applications of principles discusses. As in elasticity, attempts can be made to apply directly the virtual work principle dW = 0 in order to obtain pointwise information on the unknown solution looked for. Also more general materials may be considered like those having a non-associated flow rule. Most general principles are, however, closely connected to the method of weighted residual only.

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